Heretofore, the ordinary Wilcoxon statistic for the two-sample problem [9], [5] has been used only to test the null hypothesis that the two parent populations are identical. This paper presents a technique for utilizing the Wilcoxon statistic to test a broader type of null hypothesis, like that encountered in the Behrens-Fisher problem: we show that the usual Wilcoxon test, with $(m + n + 1)/12mn$ replaced by $1/\lbrack 4 \min(m, n)\rbrack$, may be used to test the null hypothesis of the equality of the medians of two symmetrical (continuous) distributions which are of the same form but which have different (unknown) scale parameters; more generally, the test still works for testing the equality of the medians of any two symmetrical distributions.